2-3 Relativisitic Effects in Time and Frequency
Standards
HOSOKAWA Mizuhiko
Recent technology on the precise measurement of time and frequency makes it rather
easy to detect the relativistic effects. It is, therefore, indispensable to take these effects into
account when we conduct such precise measurement. In this article, we will show the outline of the relativity and try an intimate illustration on most popular four relativistic effects;
second Doppler effect, gravitational red shift, Sagnac effect and Shapiro delay.
Keywords
Relativity, Second doppler effect, Gravitational red shift, Sagnac effect, Shapiro delay
1 Introduction―Space-Time and
the Lorentz Transformation―
One-dimensional time and three-dimensional space together constitute space-time.
Under the theory of relativity, time and space
are not independent, but instead should be
treated as a unit. Current time and frequency
standards are generated and compared with an
accuracy in the order of approximately 10-15.
However, the relativistic effects caused by the
motion of satellites or gravity near the Earth's
surface are in the order of 10-10, or over 10,000
times larger than the maximum precision of
today's devices. Such discrepancies arising in
the Newtonian concepts of space and time
have been measured and verified definitively
in a number of cases, relying on the increasing
measurement precision of current time, space,
and frequency standards. These relativistic
effects cannot be ignored in today's wide
range of high-precision time and frequency
applications.
It is now essential to take into account the
characteristics of four dimensional space-time
(specifically, relativistic effects) in the precise
treatment of time and frequency, both in global applications and in connection with today's
space-related technologies. I have discussed
the general aspects of the space-time reference
frame earlier, in the 2000 journal[1].
Four main effects should be taken into
consideration in the precise measurement of
time and frequency: the second Doppler
effect, the gravitational red shift, the Sagnac
effect, and Shapiro delay. Here, I will try to
describe these four relativistic effects as simply as possible, to provide an intuitive
overview of the problems involved and of the
concrete steps taken to address these problems.
First, I would like to clarify the basic rules
of transformation. Coordinate transformation
between two space-time reference frames in
relative motion is expressed with time and
space treated as a single quantity. In particular, when the relative velocity between the reference frames is constant and both systems
can be considered inertial systems, the applicable transformation is the well-known
Lorentz transformation. As the appendix of
Reference[1] describes the derivation of this
transformation in detail, I will provide only
the results of this transformation here, as a
basis for our treatment of relativity. Denoting
the magnitude of the relative velocity between
the reference frames as v, and assuming its
direction is along the x-axis,
HOSOKAWA Mizuhiko
17
Here, c is the velocity of light in a vacuum, 3×108 m/s, and γis the frequently used
constant, defined as follows:
version between these designations.
When space-time is expressed on twodimensional paper, one of the space axes is
chosen as the representative and is combined
with the time axis. Here, let us choose the xaxis as the representative. First, establish a
reference frame in space-time represented by a
certain time axis (t-axis) and space axis (xaxis). Next, draw a new reference frame connected to this reference frame by the Lorentz
transformation and represent it by a t'-axis and
an x'-axis. Fig.1 shows the relationship
between these two reference frames. If the
reference time clock is stationary at the origin
of the space of the first frame, it is understood
that the trajectory of this clock in space-time
(referred to as the "world line") becomes the
time axis. Then, the reference frame connected to the first reference frame by the Lorentz
transformation gains a new time axis, formed
by the clock moving at a constant velocity in
the original reference frame. It is also understood that the new space axis (x'-axis) is transformed as shown in Fig.1 due to the principle
of the constant speed of light. A given point in
space-time is designated differently in the different reference frames. The transformation
expressed in Eq.1 provides the rule for con-
When a value obtained in a reference
frame does not change in a transformation, the
value is referred to as an invariant under the
transformation. For example, in normal
Euclidean space, the components of the vector
connecting any two points change under this
rotational transformation; however, the sum of
the squares of the components (the square of
the length of the vector) remains unchanged,
in accordance with the Pythagorean theorem.
Thus, the vector length (and its square) is an
invariant in coordinate rotation. Conversely,
when one considers a transformation in which
the sum of the squares of the vector components is an invariant, rotation is obtained as
the solution. Determining precisely which
values become invariants is thus extremely
important in any attempt to define the nature
of space.
Four-dimensional space-time, in which
special relativity holds, has a unique invariant,
as described below, that is an equivalent to the
Minkowski space in mathematics. Choose
two arbitrary space-time points (t1, x1, y1, z1)
and (t2, x2, y2, z2) in an inertial reference frame.
Assume that these two points are expressed as
(t'1, x'1, y'1, z'1) and (t'2, x'2, y'2, z'2), respectively,
in another reference frame, connected to the
first by the Lorentz transformation. Under the
special theory of relativity, the values of the
space-time coordinates themselves and the differences between the components change
under the Lorentz transformation (for example, t2 –t1 ≠ t'2 –t'1). However, it can be shown
using Equation (1) that the Lorentz transformation does not change the sum of the squares
of the differences between the components:
Fig.1 Space-time and the Lorentz transformation of the coordinate axes
18
2 Invariants and the Minkowski
space: the second Doppler effect
Substituting Equation (1) in the right-hand
side of the above equation and rearranging the
Journal of the National Institute of Information and Communications Technology Vol.50 Nos.1/2 2003
terms results in the expression given on the
left-hand side of the equation (it is recommended that the reader perform this operation
once, for reference). Equation (3) holds for
any two arbitrary points. However, we will
restrict ourselves to a point (t, x, y, z) and its
vicinity (t+dt, x+dx, y+dy, z+dz). The local
characteristics of the infinitesimal line element
at an arbitrary position can then be expressed
as:
This relationship holds only for the short
period of time during which velocity may be
considered constant. In general, dt' should be
replaced by the progression of proper time dτ
Integrating Equation (5) with changing velocity gives
Equation (3') is equivalent to Equation (3)
under the special theory of relativity, except
that it is expressed in local form. Later it will
become clear that Equation (3') is a more convenient expression in terms of the general theory of relativity.
We now show that the second Doppler
effect, the time dilatation of a moving body, is
calculated using Equation (3'). Consider a
clock moving in a reference frame indicated
by numbers without dashes. The motion
could be in any direction, but for the sake of
simplicity, let us define the axes such that the
direction of motion is along the x-axis at
velocity v. The displacement during infinitesimal time dt is expressed as vdt. The vector of
motion in space-time is expressed as (dt, vdt,
0, 0). Now, choose the reference frame in
which the motion of the clock during this period appears stationary and denote it by using
numbers with dashes: in other words, choose
the reference frame that moves with the clock
as the reference frame by using numbers with
dashes. The vector of motion is now
expressed as (dt', 0, 0, 0) in this reference
frame. Here, dt' is the proper time of the moving clock.
As the value given in Equation (3') does
not change between the two reference frames,
the following is obtained:
3 Inner products and the metric
tensor
Thus, the relationship between the reading
of the moving clock and coordinate time is:
Here, the square matrices are all diagonal
matrices and their non-diagonal components
are expressed by blanks when they are 0.
Detailed discussion of this topic is more
This is known as "the second Doppler
effect" under the special theory of relativity.
Each side of Equation (3) differs in sign
from the Euclidean invariant "square length.
In Euclidean space, "square length" can be
expressed as follows, using a row vector and a
column vector:
The left-hand side of Equation (3) can be
expressed similarly as
The left-hand side of Equation (8) is
denoted as ds2. Here, ds indicates an infinitesimal line element and ds2 is its square length
in Minkowski space. With reference to Equation (8), Equation (7) can similarly be
expressed as follows:
HOSOKAWA Mizuhiko
19
appropriately reserved for mathematics textbooks; here we may simply state that the inner
product is the product of the row vector and
the transposed column vector, but only in
Euclidean space. Generally speaking, a
matrix describing the nature of the space must
be inserted between the vectors. This matrix,
which appears in the inner product of the vectors and expresses the nature of the space, is
known as the metric tensor. Tensor is the
extended notion of the vector: scalars are zeroorder tensors, and vectors are first-order tensors; tensors of higher orders can also be considered, as required. The metric tensor is a
second-order tensor and can be expressed in
equations as a square matrix. The general
expressions for higher-order tensors involve
required numbers of subscripts[2]−[4], but here
we will restrict our discussion to tensors of up
to the second order; we will express these tensors using matrices, which are intuitively easier to understand. Further, although metric tensors generally have non-zero non-diagonal
components, here we will limit our discussion
to cases in which this degree of complexity
does not arise.
The Euclidean metric tensor need not be
inserted explicitly; that is, it is expressed with
the unit matrix. On the other hand, the
Minkowski metric tensor is expressed as a
matrix with space and time having opposite
signs. Generally, the square length ds 2 of the
line element, which is an invariant under the
coordinate transformation, is expressed as follows:
ds2 = (row vector of the line element)
(metric tensor)
(column vector of the line element). (9)
These expressions may seem overly formal and of limited use. Certainly, when each
component of the metric tensor is expressed as
a constant, an expression such as Equation (3)
seems sufficient, with no need for expressions
such as Equation (8). However, using the
metric tensor provides a variety of potential
advantages in comprehending the nature of
space-time and in performing associated cal-
20
culations. Furthermore, the metric tensor is
indispensable when discussing general theory
of relativity's treatment of the curvature of
space-time. If a metric tensor is provided for
an all-encompassing space-time continuum,
the relationship between coordinate time (or
each component of coordinate time) and the
proper time or the proper length may be
obtained using this tensor. It will then be possible, for example, to compare time and frequency measurements at one position with
those at another position, or to establish a
coordinate time that can provide wide-ranging
synchronization.
The details of the metric tensor are quite
complicated, so let us concentrate on obtaining a qualitative understanding, until we establish a conclusive equation for relativistic
effects.
To say that the value of the metric tensor is
different at every point in space is equivalent
to saying that space is curved. Here it may be
helpful to consider an analogous ordinary curvature. The curvature of a given curved space
is defined by the second derivative of the metric tensor with respect to position. To say that
the space is "flat" is to say that the second
derivative of the components of the metric
tensor of the space is zero everywhere: in
other words, the metric tensor is constant
throughout the space. Further, expressing the
metric tensor as in Equation (8) means that the
space-time is flat, that all space axes are
orthogonal, uniform, and isotropic, and that
the reference frame is an inertial reference
frame that does not rotate with time.
Einstein had the insight that gravity is the
result of the warping of four-dimensional
space-time, constructing his general theory of
relativity based on this conclusion [2]−[4] .
Newton's law of universal gravitation states
that all bodies receive equal acceleration in the
gravitational field, while other theories concerning curved space (including the general
theory of relativity) state that the space-time
common to such bodies is itself curved.
The law describing the curvature of spacetime, elucidated by Einstein within the general
Journal of the National Institute of Information and Communications Technology Vol.50 Nos.1/2 2003
theory of relativity, may be qualitatively
expressed as an equation in the following
form:
curvature of space-time = distribution of
energy and momentum
(10)
It has been demonstrated in various experiments and observations that this equation
describes the laws of gravity in our world
more precisely than Newton's universal law of
gravity[5]. If one obtains the metric tensor for
all of space as the solution to Equation (10),
the practical handling of space-time may be
definitively established for applications relating to time and frequency standards.
More than one theory describes curved
space-time: apart from Einstein's general theory of relativity, the Brans-Dicke theory[5] is a
well-known example. However, observation
results to date support the general theory of
relativity. Adjusting the arbitrary parameters
of theories such as the Brans-Dicke theory can
produce results that do not contradict the general theory of relativity, but the fact remains
that no space-time phenomena have been
observed that cannot be explained based on
the general theory of relativity[5]. Thus, here
we will take only the general theory of relativity as our basic theory.
4 Approximate solution of Einstein's equation
It is extremely complicated to express
Equation (10) quantitatively, and exceedingly
difficult to obtain an exact solution. In fact,
exact solutions to this equation have been
obtained only in a few cases, each featuring
restricted conditions. However, an approximate solution is available that can supply sufficient precision for use in the weak gravitational fields found in the solar system,
although it cannot be applied to extremely
strong gravitational fields, such as those found
in the vicinity of a black hole. The expression
of the solution is not complicated for a nonrotating reference frame:
Here,Φ<0 is Newton's gravitational potential, which varies at each position in spacetime, depending on the distribution of matter.
Whether the employed reference frame is
rotating or non-rotating can be determined
mechanically based on the presence or
absence of apparent forces (such as centrifugal
force) acting on a stationary body within the
reference frame. In the reference frame of the
surface of the Earth, the centrifugal force and
Coriolis forces are observed; it can therefore
be assumed that we are dealing with a rotating
reference frame. Conversely, in a non-rotating
reference frame in which Equation (11) holds,
the Earth's surface appears to rotate while the
surface of the celestial sphere appears stationary.
The approximate solution (11) can be considered to hold up to a frequency precision of
approximately 10-17 when considering the distance between the center of the Earth and general satellite orbits, and up to approximately
10-15 when considering greater distances within
the solar system (with the exception of locations extremely close to the sun). Thus, this
approximation is capable of providing sufficient precision for current methods of generation and measurement of time and frequency
standards. Nevertheless, in consideration of
recent and anticipated improvements in the
precision of these standards, more precise
approximate solutions involving the metric
tensor have recently come under discussion [6].
In Equation (11), a position infinitely far
from all masses is chosen as the reference
point at which the metric tensor becomes
equal to the reference point for the Minkowski
space. In the Terrestrial reference frame fixed
to the Earth surface, we chose the geoid surface, a gravitational equipotential surface, as
the practical reference point for the frequency
standard. With this approach, Equation (11)
must be modified:
HOSOKAWA Mizuhiko
21
Here, Φ' is the gravitational potential with
respect to the geoid surface, and can be
expressed with potential U at the geoid surface
as follows:
It should be noted here that subtracting the
potential U at the geoid surface induces a
scale transformation on the time coordinate
axis. This scale transformation causes a difference in clock rate between Terrestrial Time
(TT), defined with respect to the proper time
on the geoid surface, and Geocentric Coordinate Time (TCG), defined with respect to the
proper time at an infinitely remote position.
The shift of the value away from 1 is
expressed by the value LG = 6.9693×10-10.
See References [1] and [4] for the definition
and transformation of these time scales.
5 Gravitational red shift
In the Terrestrial reference frame,Φ' near
the Earth's surface can be approximated using
gravitational acceleration g = 9.8 m/s 2 and
height h from the geoid surface:
With the above value for g, the correction
term 2Φ'/c 2 for the metric tensor changes by
2.2×10-16 per meter. Let us consider a stationary body near the surface of the Earth. Here,
2Φ'/c 2 << 1 holds. Further, the relation dx =
dy = dz = 0 also holds, because the position of
the body does not change after time dt. The
proper time of this body during this period is
expressed as follows:
At a position higher than the geoid surface, Φ'<0 holds; thus the higher the position
of the body, the shorter the progression of
coordinate time during the same period of
22
proper time: in other words, proper time elapses at a faster rate. It can be shown that as the
gravitational potential decreases―that is, as its
absolute value increases with respect to the
infinitely remote zero potential point of gravity in Equation (11)―time passes more slowly.
This phenomenon is known as the gravitational red shift. As Equation (14) and the value 2
Φ'/c 2 = 2.2×10-16/m indicate, the frequency for
the proper time of a body near the Earth's surface increases by 1.1×10-16 when the height of
the body increases by 1 m.
When a body is moving, the second
Doppler effect, expressed by Equation (5), is
added to this effect. This can easily be
derived by performing calculation under the
assumption that dx, dy, and dz are not zero.
Here it should be noted that as the spatial parts
of the metric tensor is divided by c2 in the
course of calculation, the effect of the shift
away from 1 is extremely small, in the order
of 1/c4. Approximating up to the order of 1/c2,
the following expression is obtained:
The proper time of a moving body can be
obtained in this manner, taking the general
theory of relativity into account in calculation.
In particular, in the case of a satellite following an elliptical orbit, velocity increases at the
perigee (where the potential is small), which
slows the proper time at the perigee even further, due to the reinforcement of the two
effects, while the reverse occurs at the apogee.
This reinforcement of the second Doppler
effect and the gravitational red shift is known
as the eccentricity effect. For circular orbits,
Equation (15) can be simplified and an orbit
referred to as a time-geostationary orbit can be
assumed for terrestrial time TT, which is
defined with respect to atomic time on the
geoid surface[7][8].
6 Rotating reference frames and
the Sagnac effect
As has been discussed, a body fixed to the
Earth's surface is rotating in a non-rotating
Journal of the National Institute of Information and Communications Technology Vol.50 Nos.1/2 2003
reference frame because the Earth is rotating.
However, almost all of the atomic clocks that
determine the standards of time and frequency
(including all of the clocks contributing to
International Atomic Time, or TAI) are fixed
to the Earth's surface. Thus when considering
the synchronization of these surface clocks,
we must take the Sagnac effect into account, a
relativistic effect that poses a number of vexing problems.
One can determine whether the reference
frame is rotating or non-rotating based on the
presence or absence of centrifugal force on a
stationary body within the reference frame.
Ignoring the effects of gravity, light in a vacuum travels at a constant velocity and in a
straight line only in a non-rotating reference
frame. In a rotating reference frame, light
travels in a helical path, depending on the
direction of the rotation. This causes the
Sagnac effect, which only occurs within a
rotating reference frame. It is best to use the
metric tensor of a rotating reference frame for
the general treatment of the Sagnac effect for
wide areas, but here we will first discuss this
effect in a local sense, using the special theory
of relativity.
Let us take a point on the equator as the
origin and consider two reference frames at a
given instant. One is a stationary reference
frame designated as Reference Frame A, and
is independent of the motion of the Earth. The
other reference frame, Reference Frame B,
moves at a constant velocity in the direction
and at the speed of the rotation of the origin at
that instant. The speed of the Earth's rotation
at the equator is approximately 450 m/s. The
point chosen as the origin at the given instant
moves in Reference Frame A at this speed of
rotation, but can be considered to remain at
the origin in Reference Frame B for a short
period of time, during which the effect of rotation is negligible. As shown in Fig.2 and in
the Lorentz transformation (1), the two times
for the same space-time point diverge as the
distance from the origin in Reference Frame A
increases. Obviously Reference Frame A can
more broadly express time and space on the
Earth, because the Earth is rotating with
respect to the inertial reference frame. The
preparation and linkage of additional reference
frames (such as Reference Frame B) for multiple points on the equator will result in multiple time values―i.e., a different time for each
reference frame (Fig.3).
Fig.2 Inertial reference frame and a refer-
ence frame that follows the rotation
of the Earth
Fig.3 Linkage of reference frames following
the rotation of the Earth
If these frames were to be linked and time
shifted to force consistency among the frames,
time after one rotation will differ significantly
from the original time. It is clear that there is
no way of preparing rotating reference frames
that are both locally inertial and capable of
sharing common time over a wide range. This
is due to the phenomenon in which the speed
HOSOKAWA Mizuhiko
23
of eastbound light decreases by the speed of
rotation and the speed of westbound light
increases by the same amount within a reference frame moving along with the rotation of
the Earth. This phenomenon is a logical consequence of the constant velocity of light
within a non-rotating reference frame (independent of the eastbound rotation of the
Earth). This effect is referred to as the Sagnac
effect, after the scientist who discovered it.
Reference [9] is a well-known work describing
the appearance of this effect in international
time comparison.
The Sagnac effect poses a significant
problem in the establishment of time standards
using instruments fixed on the surface of the
Earth. Two methods can be used to resolve
this problem: the definition of the propagation
velocity of light could be modified to allow
for different speeds for eastbound light and
westbound light, or a correction term could be
introduced to describe the time delay for a
point in the east relative to another point in the
west. The latter method is employed in practice. The correction term Δt for distance x in
the east-west direction is expressed as follows:
Here, ΩE is the angular velocity of the
Earth, R is the radius of the Earth, x/c is the
time required for light to travel distance x,ΩE
Rx/c is the distance the Earth surface moves
during this period of time, and Δt is the time
required for light to catch up. A more general
treatment of rotating reference frames in time
comparison using the metric tensor is
described in detail in Reference[10]. The metric tensor of a rotating reference frame has
non-zero non-diagonal components; accordingly its treatment is more complicated.
7 Shapiro delay
In the theory of curved space-time, the
motion of light in a vacuum is determined
based on the assumption that the length given
in Equation (11) for the line element ds (the
motion of light) of the world line is zero
24
everywhere. This corresponds to the conclusion that space can be locally approximated as
Minkowski space and that a coordinate transformation may be obtained in which the special theory of relativity holds. This in turn
indicates that the special theory of relativity
does not always hold globally and that light
does not always travel at a constant speed in
curved space-time. Let us consider the simple
example of light traveling along the x-axis
with a mass M at the origin, using the approximated solution (11) of the general theory of
relativity. Here, y = z = 0 and Φ =–GM/x
always hold; thus the metric tensor is simplified to a 2×2 matrix:
Here, G is Newton's gravitational constant,
6.67×10-11 m3/(kgs2). The motion of light is
obtained by expanding this expression and
substituting ds2 = 0:
From this equation and the approximation
1 >> 2GM/xc 2 for a weak gravitational field,
the velocity of light dx/dt at point (t, x) in
space-time within this reference frame is
smaller than the so-called velocity of light c:
This equation clearly shows that the velocity of light is zero at x = 2GM/c 2, which is separated from the origin by the Schwarzschild
radius; in this case the distance from the origin
is small and the gravitational field is strong.
The equation also shows that the velocity of
light equals the normal value c at the far limit,
where x is large. Note also that velocity does
not depend on the direction of the motion of
light; i.e., on whether it is traveling toward or
away from mass M, the source of the gravitational field.
As demonstrated in this example, in a reference frame that satisfies the general theory
of relativity in a wide range, the velocity of
light becomes smaller where the gravitational
field is stronger, and as a result, it is deduced
Journal of the National Institute of Information and Communications Technology Vol.50 Nos.1/2 2003
that the propagation time of light within the
gravitational field will appear to decrease relative to that obtained assuming velocity c. I.
Shapiro first described this effect in a paper
submitted to a referred journal[5][11]. This
effect is thus referred to as Shapiro delay. In
his paper, Shapiro demonstrated that the propagation of light or electromagnetic waves
between planets shows delay when calculated
within the framework of the general theory of
relativity, relative to results of calculation that
do not take the general theory of relativity into
account. Until then, only three experiments
had been proposed to verify the general theory
of relativity: one dealing with the movement
of perihelion of Mercury, the gravitational lens
effect, and the gravitational red shift. Thus
Shapiro entitled his paper the "Fourth Test of
the General Theory of Relativity." This effect
was observed in radar-echo experiments
involving Mercury and Venus and played a
significant role in the verification of the general theory of relativity[5].
The delay in propagation time can easily
be calculated, using Equation (19), on an axis
that passes through the gravity source, as in
the example above. Considering the two
points x1 and x2 in the positive region on the xaxis, the definite integral of
yields delayΔt during actual propagation
of light from x1 to x2, relative to propagation
assuming velocity c:
For general propagation in three-dimensional space using a post-Galilei's approximation of sufficient precision, based on the
length of the sides of the triangle made by x1,
x2, and the position of the gravity source P
(Fig.4),Δt can be obtained by solving the following expression:
As an example, let us assume that x1 is the
radius of revolution of Venus (0.7 astronomical units), x2 is the radius of revolution of the
Fig.4 Shapiro delay: triangle made by the
initial and final points and the gravity
source
Earth (1 astronomical unit), and M is the mass
of the Sun, 2×1030 kg. In this case the Shapiro
delay for the propagation of light between
Venus and the Earth at the inferior conjunction
of Venus when the Sun, Venus, and the Earth
are arranged in a line is approximately 3.5
microseconds one way, and 7 microseconds
for the round trip.
The elongation of the light path attributable to bending due to the gravitational lens
effect is negligible compared to the Shapiro
delay, as long as the bending angle is not
large. As the bending angle resulting from the
gravitational lens effect of the Sun is less than
2 arc-seconds (even where this effect is greatest, where the light grazes the edge of the sun)
such bending may be deemed negligible within the solar system.
8 Conclusion
Using the Lorentz transformation, we have
introduced the metric tensor as an invariant
and have outlined four relativistic effects: the
second Doppler effect, the gravitational red
shift, the Sagnac effect, and Shapiro delay.
These effects all seem strange at first, but
become intuitive and easy to understand once
HOSOKAWA Mizuhiko
25
the concept of space-time is grasped. In this
sense, I hope that this article will prove of
some assistance to the reader's understanding
of the nature of space-time. I could not present a number of more detailed treatments and
more general expressions here; please see the
books listed in the reference list for further
information. I hope to provide more detailed
quantitative treatments and descriptions, in
later reports. In the meantime, the reader is
recommended to consult available articles on
the influence of relativistic effects on the GPS
[12], which provide useful discussions of the
subject at hand.
I hope that this report will provide an intuitive understanding of relativistic concepts and
of the four effects discussed above, and that it
will assist in the evaluation of the type and
amount of practical effects, particularly in
activities such as time and frequency measurement or time comparison.
References
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2 A.Einstein, S. B. Preuss.Akad. Wiss., pp.777-786, pp.799-801 , 1915.
3 L.D.Landau, E.M.Lifshiz, "The Classical Theory of Fields" §6, Butterworth-Heinemann; 4th edition. 1987.
4 T.Hukushima, "Gendai Sokuchigaku", §3, pp.105-154, Bumkenshya, 1994. (in Japanese)
5 Clifford M.Will, "Was Einstein Right? Putting General Relativity to the Test", Second edition published by
Basic Books, New York, 1993.
6 IAU Colloquium 180 Resolutions, 30 March 2000. http://danof.obspm.fr/IAU_resolutions/Resol-UAI.htm
7 M.Hosokawa and F.Takahashi, Publ. Astron. Soc. Japan 44, pp.159-162 , 1992.
8 M.Hosokawa, F.Takahashi, and M.Yoshikawa, Review of the Communications Research Laboratory, Time
scales and Frequency Standards Special issue, Vol.45,, Nos.1/2, pp.103-, 1999. (in Japanese)
9 Y.Saburi, M.Yamamoto, and K.Harada, IEEE Trans. Instrum. Meas., IM-25 No4, pp.473 - 477, 1976.
10 G.Petit and P.Wolf, Astron.Astrophys. 286, pp.971-977, 1994.
11 I. Shapiro, Phys. Rev. Lett. 13, pp.789-, 1964.
12 N.Ashby, M.Hosokawa, Parity Jun. pp.18-, Maruzen Co.Ltd., Tokyo, 2003.
HOSOKAWA Mizuhiko, Ph. D.
Leader, Atomic Frequency Standards
Group, Applied Research and Standards Division
Atomic Frequency Standards, Space
Time Measurements
26
Journal of the National Institute of Information and Communications Technology Vol.50 Nos.1/2 2003